Problem : Safety zone around playground swings Children or objects that children have with them (e.g. sweets, trainers, purse) may well fall off a playground swing when it is in motion. In fear of possible litigation, a town council is proposing to establish a soft landing zone around a swing, so that if a child or an object does fall off it, then any consequences of such an accident are reduced. Advise the council on the minimum area that should be established as a soft landing zone around a playground swing.

Specify the purpose • Define the specific problem to be solved. Write a clear, succinct statement of the specific problem addressed in your report, in your own words (do not just repeat the specification on page 3). • Describe the features that you are going to investigate. Give some indication of the approach used to create the model.

Create the model • Outline the mathematics to be used in the model. Give a qualitative description of the approach to be used in the first model, to explain why and how the first model will be formulated.

• State assumptions. Create a numbered list of clearly stated assumptions used in the model (take care not to miss assumptions or include assumptions that are never used). Do not attempt to justify assumptions here. Data values should not appear in assumptions, so, for example, 'the width of the road is 10 m' should be replaced by 'the width of the road is constant'.

• Choose variables and parameters. Create a table of all symbols used in the model. For each symbol, state a clear definition and its associated units. It is not necessary to distinguish between variables and parameters.

• Formulate mathematical relationships. Derive relationships between your variables and parameters. You should explain how the equations follow from your assumptions, which should be referenced. Do the mathematics

• Derive a first model. Solve your first model to find the variable of interest (as specified in the purpose of the model) in terms of other Variables and parameters. Clearly state the mathematical model derived. It is not necessary to have one overall explicit equation; it is possible to have a series of equations, which may aid clarity, or an implicit equation (that will be solved numerically). Your solution at this stage should not include particular data values.

• Draw graphs showing typical relationships. Sketch graphs to show the expected variation of variables predicted by your model. Use typical values for any parameters.

• Check your model using dimensional analysis. Interpret the results

• Collect relevant data for parameter values. Usually relevant data is available on the internet (or in the library), in which case the source should be referenced. For some models it is easier to perform a simple experiment, in which case the deduced parameter values should be stated here and the experiment should be described in an appendix.

• Describe the mathematical solution. Substitute data into your model to find a solution. Clearly state in words this solution and how it relates to the purpose of the model. This should be written in a form that could be understood by a lay-person, by presenting it, for example, as a set of instructions, a graph or a table.

• Find predictions to compare with reality. Look for any predictions of your model, or part of your model, or a corollary of your model that may be tested.

Evaluate the model • Collect data to compare with the model. Collect additional data to test your model. Do not use the data used to define parameter values. As before, the additional data can be from the internet, the library or an experiment.

• Test your first model. Compare model predictions with the additional data that you collected. Some models may be impossible to test in this way, in which case you should explain why it is not possible to test your model. Some marks are available for describing an experiment without actually being able to perform it. • Criticise your first model. Criticise your model based on the tests that you performed.

• Review your assumptions. Consider each assumption in turn, and explain what would be the effect of changing it – would it improve the model to fit better with the evaluation? Revise the model

• Decide whether to revise your first model. Decide whether a revision of your first model is justified. Explain why you made your decision, referring to the evaluation of the first model and your review of the assumptions. If your first model fits your data well, then consider if a simpler model might be better.

• Describe your intended revision. Include a clear statement of any assumptions that are being revised and the new assumption(s) that will replace them. Note that a change of a parameter value does not constitute a revision of the model.

Conclusions • Summarise your modelling. Include the performance of your first model, any attempts to improve on it, and any comments on the modelling process. This short summary should not introduce any new considerations.